Easy Construction of Bicentric Quadrilateral:
What Is This About?
A Mathematical Droodle
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Copyright © 19962018 Alexander Bogomolny
Easy Construction of Bicentric Quadrilateral
A quadrilateral is bicentric if it's both inscriptable and circumscriptable. (Inscriptable means admitting an incircle. Circumscriptable means cyclic, i.e., admitting a circumcircle.) Bicentric quadrilaterals might seem exotic, but the applet shows how such quadrilaterals can be constructed easily. (Another construction appears elsewhere.)
What if applet does not run? 
Let ABCD be a cyclic quadrilateral with vertices on a given circle w. Assume ABCD is also orthodiagonal, i.e.,
The latter assertion can be rephrased. We know that, the diagonals of a cyclic quadrilateral (ABCD in this case) and those of the quadrilateral formed by the points of tangency of ABCD and its incircle, are concurrent. Therefore we can say that, for a bicentric quadrilateral, the intersecition of the diagonals M, the incenter I and the circumcenter O are collinear.
Proof
First of all note the following angle identities
(1)  ∠QCB = ∠BCQ = ∠BAC = ∠BDC = α. 
(2)  ∠ADS = ∠DAS = ∠ABD = ∠ACD = β. 
(3)  ∠AEB = ∠CED = γ. 
We have to show that the diagonals of ABCD are orthogonal, i.e.,
∠PQR + ∠PSR = 180°, 
or, which is the same, iff
(4)  ∠BQC + ∠ASD = 180° 
Now, in ΔBCQ,
∠BQC + 2α = 180°, 
so that from (1)
(*_{1})  ∠BQC + 2∠BAC = 180°, 
In ΔADS,
∠ASD + 2β = 180°, 
so that from (2)
(*_{2})  ∠ASD + 2∠ABD = 180°, 
Thus, in ΔABE,

which proves the first part of the assertion. The fact that I, O, and E are collinear has been proven in [Dubrovsky] and [Honsberger]. An absolutely delicious proof appears as a consequence of another construction of bicentric quadrilaterals.
References
 G. Bennett, Bicentric Quadrilaterals and the Pedal ngon, in The Lighter Side of Mathematics, R.K.Guy and R.E.Woodrow, eds, MAA, 1994, p. 97
 J. L. Coolidge, A Treatise On the Circle and the Sphere, AMS  Chelsea Publishing, 1971, p. 45
 H. Dorrie, 100 Great Problems Of Elementary Mathematics, Dover Publications, NY, 1965, pp. 188193
 V. N. Dubrovsky, Solution to problem M1154, Kvant, n 8, 1989, pp 3435 (in Russian), pdf is available at https://kvant.mccme.ru/1989/08/p34.htm.
 R. A. Johnson, Advanced Euclidean Geometry (Modern Geometry), Dover, 1960, p. 95
 R. Honsberger, In Pólya's Footsteps, MAA, 1999, pp. 100101
Bicentric Quadrilateral
 Collinearity in Bicentric Quadrilaterals
 Easy Construction of Bicentric Quadrilateral
 Easy Construction of Bicentric Quadrilateral II
 Projective Collinearity in a Quadrilateral
 Fuss' Theorem
 Line IO in Bicentric Quadrilaterals
 Area of a Bicentric Quadrilateral
 Concyclic Incenters in Bicentric Quadrilateral
Activities Contact Front page Contents Geometry Eye opener
Copyright © 19962018 Alexander Bogomolny
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